document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=79946>Question 79946</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. If y = a(x - 3)^2 + c and y = (2x - 3)(2x + b) represent the same quadratic function, find a, b, and c. (There are two possibles methods: i) expand both or ii) use the symmetry of the roots around the vertex.) \r<BR>\n" );
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document.write( "- I expanded both equations but I could not simplify the terms.\r<BR>\n" );
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document.write( "Hello there,\r<BR>\n" );
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document.write( "   Thank you in advance for your help. For the following problem, I was able to solve it using method 2, but I do not know how to solve it using method 1.\r<BR>\n" );
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document.write( "Best,<BR>\n" );
document.write( "James </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #57330 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=57330\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+a%28x-3%29%5E2%2Bc ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+a%28x-3%29%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%282x-3%29%282x%2Bb%29 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%282x-3%29%282x%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Expand both equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+a%28x%5E2-6x%2B9%29%2Bc ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+a%28x%5E2-6x%2B9%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+ax%5E2-6ax%2B%289a%2Bc%29 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+ax%5E2-6ax%2B%289a%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+4x%5E2%2B%282b-6%29x-3b ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+4x%5E2%2B%282b-6%29x-3b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since these two equations represent the same quadratic function, the corresponding terms should be equal, so comparing term-for-term,...<BR>\n" );
document.write( "First term:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+=+4x%5E2 ALIGN=MIDDLE  ALT=\"ax%5E2+=+4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+4 ALIGN=MIDDLE  ALT=\"a+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Second term:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6ax+=+%282b-6%29x ALIGN=MIDDLE  ALT=\"-6ax+=+%282b-6%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6a+=+2b-6 ALIGN=MIDDLE  ALT=\"-6a+=+2b-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute a = 4.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%284%29+=+2b-6 ALIGN=MIDDLE  ALT=\"-6%284%29+=+2b-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24+=+2b-6 ALIGN=MIDDLE  ALT=\"-24+=+2b-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b+=+-18 ALIGN=MIDDLE  ALT=\"2b+=+-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-9 ALIGN=MIDDLE  ALT=\"b+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Third term:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a%2Bc+=+-3b ALIGN=MIDDLE  ALT=\"9a%2Bc+=+-3b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute a = 4 and b = -9<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%284%29%2Bc+=+-3%28-9%29 ALIGN=MIDDLE  ALT=\"9%284%29%2Bc+=+-3%28-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2Bc+=+27 ALIGN=MIDDLE  ALT=\"36%2Bc+=+27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+-9 ALIGN=MIDDLE  ALT=\"c+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Check: Substitute a = 4, b = -9, and c = -9<BR>\n" );
document.write( "First equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+4%28x-3%29%5E2%2B%28-9%29 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+4%28x-3%29%5E2%2B%28-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+4%28x%5E2-6x%2B9%29%2B%28-9%29 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+4%28x%5E2-6x%2B9%29%2B%28-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+4x%5E2-24x%2B%2836%2B%28-9%29%29 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+4x%5E2-24x%2B%2836%2B%28-9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+4x%5E2-24x%2B27 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+4x%5E2-24x%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Second equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%282x-3%29%282x%2B%28-9%29%29 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%282x-3%29%282x%2B%28-9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%282x-3%29%282x-9%29 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%282x-3%29%282x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+4x%5E2-18x-6x%2B27 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+4x%5E2-18x-6x%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+4x%5E2-24x%2B27 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+4x%5E2-24x%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As you can see, the two equations are identical if:<BR>\n" );
document.write( "a = 4, b = -9, and c = -9\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );